Which structure of the inner ear measures sound waves and therefore is involved with hearing?

The correct answer is - Cochlea.

The snail-shell-like cochlea is one of the two parts of the inner ear, and this the part that is involved with the hearing by measuring the sound waves that are delivered to it through the ossicular chain that connects it with the tympanic membrane, which further gets the sound waves from the outer ear. It is a highly specialized sensory organ that is connected with the eight cranial nerve.

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